3.30.88 \(\int \frac {\sqrt {-b+a^2 x^2} (d+c x^4) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx\)

Optimal. Leaf size=390 \[ \frac {a \sqrt [4]{b} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {a^2 x^2-b}+a x-\sqrt {b}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \tanh ^{-1}\left (\frac {\frac {\sqrt {a^2 x^2-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt {\sqrt {a^2 x^2-b}+a x}}\right )}{\sqrt {2}}+\frac {224 a^9 c x^{10}+2016 a^9 d x^6-504 a^7 b c x^8-2016 a^7 b d x^4+126 a^5 b^2 c x^6+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4+63 a^3 b^3 d+\sqrt {a^2 x^2-b} \left (224 a^8 c x^9+2016 a^8 d x^5-392 a^6 b c x^7-1008 a^6 b d x^3-42 a^4 b^2 c x^5-126 a^4 b^2 d x+154 a^2 b^3 c x^3-16 b^4 c x\right )-72 a b^4 c x^2}{63 a^3 x \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}} \]

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Rubi [A]  time = 1.97, antiderivative size = 433, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 14, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6742, 2120, 463, 12, 321, 329, 211, 1165, 628, 1162, 617, 204, 259, 270} \begin {gather*} 2 a d \sqrt {\sqrt {a^2 x^2-b}+a x}+\frac {2 a b d \sqrt {\sqrt {a^2 x^2-b}+a x}}{\left (\sqrt {a^2 x^2-b}+a x\right )^2+b}+\frac {a \sqrt [4]{b} d \log \left (\sqrt {a^2 x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{2 \sqrt {2}}-\frac {a \sqrt [4]{b} d \log \left (\sqrt {a^2 x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{2 \sqrt {2}}+\frac {a \sqrt [4]{b} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2}}-\frac {b^4 c}{56 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/2}}-\frac {b^2 c \sqrt {\sqrt {a^2 x^2-b}+a x}}{4 a^3}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}}{72 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-b + a^2*x^2]*(d + c*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/x^2,x]

[Out]

-1/56*(b^4*c)/(a^3*(a*x + Sqrt[-b + a^2*x^2])^(7/2)) - (b^2*c*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/(4*a^3) + 2*a*d*
Sqrt[a*x + Sqrt[-b + a^2*x^2]] + (c*(a*x + Sqrt[-b + a^2*x^2])^(9/2))/(72*a^3) + (2*a*b*d*Sqrt[a*x + Sqrt[-b +
a^2*x^2]])/(b + (a*x + Sqrt[-b + a^2*x^2])^2) + (a*b^(1/4)*d*ArcTan[1 - (Sqrt[2]*Sqrt[a*x + Sqrt[-b + a^2*x^2
]])/b^(1/4)])/Sqrt[2] - (a*b^(1/4)*d*ArcTan[1 + (Sqrt[2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/4)])/Sqrt[2] + (
a*b^(1/4)*d*Log[Sqrt[b] + a*x + Sqrt[-b + a^2*x^2] - Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(2*Sqrt[
2]) - (a*b^(1/4)*d*Log[Sqrt[b] + a*x + Sqrt[-b + a^2*x^2] + Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(
2*Sqrt[2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 259

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[(c*x)
^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (Intege
rQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 463

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> -Simp[((b*c - a*
d)^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b^2*e*n*(p + 1)), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a + b
*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a,
b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2120

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1*(i/c)^m)/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)), Subst[Int[x^(n - 2*m - p - 2)*(-(a*f^2) + x^2)^p*(a*f^2
+ x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0
] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx &=\int \left (\frac {d \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2}+c x^2 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \, dx\\ &=c \int x^2 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \, dx+d \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx\\ &=\frac {c \operatorname {Subst}\left (\int \frac {\left (-b+x^2\right )^2 \left (b+x^2\right )^2}{x^{9/2}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{16 a^3}+(a d) \operatorname {Subst}\left (\int \frac {\left (-b+x^2\right )^2}{\sqrt {x} \left (b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {c \operatorname {Subst}\left (\int \frac {\left (-b^2+x^4\right )^2}{x^{9/2}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{16 a^3}-\frac {(a d) \operatorname {Subst}\left (\int -\frac {2 b x^{3/2}}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 b}\\ &=\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {c \operatorname {Subst}\left (\int \left (\frac {b^4}{x^{9/2}}-\frac {2 b^2}{\sqrt {x}}+x^{7/2}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{16 a^3}+(a d) \operatorname {Subst}\left (\int \frac {x^{3/2}}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=-\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}-(a b d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=-\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}-(2 a b d) \operatorname {Subst}\left (\int \frac {1}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}-\left (a \sqrt {b} d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-\left (a \sqrt {b} d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {\left (a \sqrt [4]{b} d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}+\frac {\left (a \sqrt [4]{b} d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {1}{2} \left (a \sqrt {b} d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{2} \left (a \sqrt {b} d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {\left (a \sqrt [4]{b} d\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}}+\frac {\left (a \sqrt [4]{b} d\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}}\\ &=-\frac {b^4 c}{56 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}-\frac {b^2 c \sqrt {a x+\sqrt {-b+a^2 x^2}}}{4 a^3}+2 a d \sqrt {a x+\sqrt {-b+a^2 x^2}}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}{72 a^3}+\frac {2 a b d \sqrt {a x+\sqrt {-b+a^2 x^2}}}{b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2}+\frac {a \sqrt [4]{b} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2}}+\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}-\frac {a \sqrt [4]{b} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [B]  time = 23.41, size = 11755, normalized size = 30.14 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[-b + a^2*x^2]*(d + c*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/x^2,x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.89, size = 390, normalized size = 1.00 \begin {gather*} \frac {63 a^3 b^3 d-72 a b^4 c x^2+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4-2016 a^7 b d x^4+126 a^5 b^2 c x^6+2016 a^9 d x^6-504 a^7 b c x^8+224 a^9 c x^{10}+\sqrt {-b+a^2 x^2} \left (-16 b^4 c x-126 a^4 b^2 d x+154 a^2 b^3 c x^3-1008 a^6 b d x^3-42 a^4 b^2 c x^5+2016 a^8 d x^5-392 a^6 b c x^7+224 a^8 c x^9\right )}{63 a^3 x \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\frac {a \sqrt [4]{b} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*(d + c*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/x^2,x]

[Out]

(63*a^3*b^3*d - 72*a*b^4*c*x^2 + 126*a^5*b^2*d*x^2 + 210*a^3*b^3*c*x^4 - 2016*a^7*b*d*x^4 + 126*a^5*b^2*c*x^6
+ 2016*a^9*d*x^6 - 504*a^7*b*c*x^8 + 224*a^9*c*x^10 + Sqrt[-b + a^2*x^2]*(-16*b^4*c*x - 126*a^4*b^2*d*x + 154*
a^2*b^3*c*x^3 - 1008*a^6*b*d*x^3 - 42*a^4*b^2*c*x^5 + 2016*a^8*d*x^5 - 392*a^6*b*c*x^7 + 224*a^8*c*x^9))/(63*a
^3*x*(a*x + Sqrt[-b + a^2*x^2])^(9/2)) + (a*b^(1/4)*d*ArcTan[(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/
(-Sqrt[b] + a*x + Sqrt[-b + a^2*x^2])])/Sqrt[2] - (a*b^(1/4)*d*ArcTanh[(b^(1/4)/Sqrt[2] + (a*x)/(Sqrt[2]*b^(1/
4)) + Sqrt[-b + a^2*x^2]/(Sqrt[2]*b^(1/4)))/Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/Sqrt[2]

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fricas [A]  time = 0.73, size = 329, normalized size = 0.84 \begin {gather*} -\frac {252 \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \arctan \left (-\frac {\left (-a^{4} b d^{4}\right )^{\frac {3}{4}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d - \left (-a^{4} b d^{4}\right )^{\frac {3}{4}} \sqrt {a^{3} d^{2} x + \sqrt {a^{2} x^{2} - b} a^{2} d^{2} + \sqrt {-a^{4} b d^{4}}}}{a^{4} b d^{4}}\right ) + 63 \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d + \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) - 63 \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d - \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) + 2 \, {\left (2 \, a^{4} c x^{5} - 2 \, a^{2} b c x^{3} - {\left (189 \, a^{4} d - 16 \, b^{2} c\right )} x - {\left (16 \, a^{3} c x^{4} - 8 \, a b c x^{2} - 63 \, a^{3} d\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{126 \, a^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x, algorithm="fricas")

[Out]

-1/126*(252*(-a^4*b*d^4)^(1/4)*a^3*x*arctan(-((-a^4*b*d^4)^(3/4)*sqrt(a*x + sqrt(a^2*x^2 - b))*a*d - (-a^4*b*d
^4)^(3/4)*sqrt(a^3*d^2*x + sqrt(a^2*x^2 - b)*a^2*d^2 + sqrt(-a^4*b*d^4)))/(a^4*b*d^4)) + 63*(-a^4*b*d^4)^(1/4)
*a^3*x*log(sqrt(a*x + sqrt(a^2*x^2 - b))*a*d + (-a^4*b*d^4)^(1/4)) - 63*(-a^4*b*d^4)^(1/4)*a^3*x*log(sqrt(a*x
+ sqrt(a^2*x^2 - b))*a*d - (-a^4*b*d^4)^(1/4)) + 2*(2*a^4*c*x^5 - 2*a^2*b*c*x^3 - (189*a^4*d - 16*b^2*c)*x - (
16*a^3*c*x^4 - 8*a*b*c*x^2 - 63*a^3*d)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)))/(a^3*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x, algorithm="giac")

[Out]

integrate((c*x^4 + d)*sqrt(a^2*x^2 - b)*sqrt(a*x + sqrt(a^2*x^2 - b))/x^2, x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (c \,x^{4}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x)

[Out]

int((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x, algorithm="maxima")

[Out]

integrate((c*x^4 + d)*sqrt(a^2*x^2 - b)*sqrt(a*x + sqrt(a^2*x^2 - b))/x^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,\left (c\,x^4+d\right )\,\sqrt {a^2\,x^2-b}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x + (a^2*x^2 - b)^(1/2))^(1/2)*(d + c*x^4)*(a^2*x^2 - b)^(1/2))/x^2,x)

[Out]

int(((a*x + (a^2*x^2 - b)^(1/2))^(1/2)*(d + c*x^4)*(a^2*x^2 - b)^(1/2))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b} \left (c x^{4} + d\right )}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**2-b)**(1/2)*(c*x**4+d)*(a*x+(a**2*x**2-b)**(1/2))**(1/2)/x**2,x)

[Out]

Integral(sqrt(a*x + sqrt(a**2*x**2 - b))*sqrt(a**2*x**2 - b)*(c*x**4 + d)/x**2, x)

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